Reference and Synthesis
Common Algebraic Errors and How to Avoid Them
The errors that recur across every topic in this collection, why each is tempting, and a counterexample or check that settles each one in seconds.
What this page covers
- Recognise the standard error patterns before making them
- Apply a numerical test to settle a doubtful step
- Understand why each error is tempting
- Build habits that prevent whole classes of mistake
Assuming something distributes
Multiplication distributes over addition. Almost nothing else does, and this single over-generalisation accounts for more errors than any other cause.
| Wrong | Right | Counterexample |
|---|---|---|
| (a + b)2 = a2 + b2 | a2 + 2ab + b2 | a = b = 1: 4, not 2 |
| √a + b = √a + √b | No simplification | √9 + 16 = 5, not 7 |
| 1a + b = 1a + 1b | aba+b inverted | a = b = 1: 12, not 2 |
| sin(u + v) = sin u + sin v | sin ucos v + cos usin v | u = v = π2: 0, not 2 |
| |a + b| = |a| + |b| | ≤, with equality only for matching signs | |5 - 7| = 2, not 12 |
| (ab)n = an + bn | anbn | a = b = 2, n = 2: 16, not 8 |
Whenever tempted to split an operation across a sum, substitute a = b = 1 and compare. It takes three seconds and settles every row above.
Sign errors
| Wrong | Right | Why it happens |
|---|---|---|
| -(a - b) = -a - b | -a + b | The minus was applied to one term only |
| 1 - (-3) = -2 | = 4 | Double negative not resolved |
| (-x)2 = -x2 | = x2 | The bracket was ignored |
| b2 = -25 when b = -5 | = 25 | The square of a negative is positive |
| -4ac with c < 0 taken as negative | -4(2)(-2) = +16 | Two negatives were not combined |
| (5i)2 = 25 | = -25 | i2 = -1 was forgotten |
Whenever a minus sign stands in front of a bracket, expand it on a separate line before doing anything else. The extra line costs seconds and eliminates the entire category.
Cancelling what cannot be cancelled
Cancellation removes a factor common to the whole numerator and the whole denominator. It never removes a term.
| Wrong | Right | Test |
|---|---|---|
| x + 2x = 2 | No simplification | At x = 2 it is 2; at x = 1 it is 3 |
| x2 + 1x + 1 = x | No simplification | At x = 1: 22 = 1, not 1... and at x = 2: 53, not 2 |
| a + ba = b | 1 + ba | The numerator is a sum |
Ask: can the numerator be written as (the thing to cancel) times something? If not, nothing cancels.
Losing or gaining solutions
| Operation | Risk | Remedy |
|---|---|---|
| Dividing by a variable | Loses solutions | Factor instead: x2 = 3x gives x(x - 3) = 0 |
| Squaring both sides | Gains solutions | Check every candidate in the original |
| Taking a square root | Loses one | Include the ± |
| Multiplying by an expression | May gain | Check the multiplier is non-zero |
| Cancelling a common factor from an equation | Loses | Set the factor to zero as a separate case |
The first two are the important ones. Dividing x2 = 3x by x discards x = 0; squaring √x + 2 = x - 4 introduces x = 2, which fails on substitution.
Even powers and even roots
| Wrong | Right | Test |
|---|---|---|
| √x2 = x | |x| | At x = -3: 3, not -3 |
| (x6)1/2 = x3 | |x3| | At x = -2: 8, not -8 |
| x2 = 9 gives x = 3 | x = ±3 | Both squares give 9 |
| (-16)1/4 = -2 | Not a real number | (-2)4 = 16 |
| √4 = ±2 | = 2 | The symbol names the principal root |
√4 = 2 (one value), but x2 = 4 gives x = ±2 (two values). The symbol names a number; the equation asks which numbers satisfy it.
Index law errors
| Wrong | Right | Counterexample |
|---|---|---|
| aman = amn | am+n | 22 · 23 = 32, not 64 |
| (am)n = am+n | amn | (22)3 = 64, not 32 |
| a0 = 0 | a0 = 1 | Otherwise Law 1 fails |
| a-m = -am | 1am | 2-3 = 18 |
| 4y-2 = 14y2 | 4y2 | The exponent attaches to y only |
| 23 · 34 = 67 | No simplification | Different bases do not combine |
Errors specific to the later topics
| Wrong | Right | Note |
|---|---|---|
| cos(u+v) = cos ucos v + sin usin v | The sign is minus | Check at u = v = π2: must give -1 |
| x = sin t, y = cos t | The other way round | At t = 0 the point is (1, 0) |
| sec t = 1sin t | 1cos t | The co in cosecant pairs it with sine |
| AB = BA for matrices | Generally false | Compute both and compare |
| a21 read as row 1, column 2 | Row index first | a21 is row 2, column 1 |
| Adding vector magnitudes | Add components | |u + v| ≠ |u| + |v| |
| Multiplying out a variable denominator in an inequality | Use a sign diagram | Its sign may reverse the inequality |
Three habits worth building
- Substitute a number. Test any doubtful identity at x = 1 or x = 2. Agreement is not proof; disagreement is refutation. This one habit catches most of the errors above.
- Expand a leading minus on its own line. Never distribute a negative sign mentally. It costs one line and removes a whole category of error.
- Ask what could be zero. Before dividing by anything containing a variable, before cancelling, and whenever a denominator appears. Every excluded value should be recorded when it arises, not recovered later.
A fourth is worth adding for longer work: check in the original, never in a line partway through. An error made early will be faithfully reproduced by any check that starts downstream of it.
Frequently asked questions
What is the single most useful check?
Substituting a convenient number. Most errors on this page are exposed in seconds by testing at x = 1 or x = 2. Agreement is not proof, but disagreement is instant refutation.
Why do these errors keep happening?
Almost all of them are over-generalisations of a rule that is true somewhere else. Multiplication distributes over addition, so it feels as though squaring should too.
Is there a pattern to them?
Two: assuming an operation distributes when it does not, and losing a sign when a negative is involved. Between them they account for most of the list.
How do I stop making them?
Slow down at exactly three points: any minus sign in front of a bracket, any division by something containing a variable, and any even root or even power.
Source. Compiled from the whole source, and from the pitfalls the teaching notes flag explicitly.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.